By S. R. Caradus

Publication through S. R. Caradus, W. E. Pfaffenberger

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Is a Riesz operator, then by the above decomposition lemma, the non zero points of sp(T) are poles. Moreover, since is the range of the spectral projection corresponding to N [(X^I - T)^] ^ ^ and, by assumption, this is a finite dimensional space, each pole has finite multiplicity. 2 that X^ X I - T Then finite dimensional, N[(X I - T)^] = R(P ) O R[(X I - T)^] 0 so that Ф(Т) = Й - {0} and hence 0 Similarly, O <«. 0 T p denote R[(X I - T)^] = N(P ) O has finite codimension. ß(XI-T) Let and since the latter is O a(X I - T) < «.

The reader who has been exposed to the theory of Banach algebras should certainly skip ahead. An algebra A space in the norm ||xy|| ^ ||x|| over the complex (real) field which is a normed linear I|x|| ||y|| is called a complex (real) normed algebra if for all it is called a Banach algebra. x,y in A. If A is complete in this norm We shall confine our attention to the complex case, but point out that the theory of real normed Banach algebras can be reduced to the complex case [54]. For our later work the main examples of Banach algebras are derived from the algebra X B(X) of all bounded linear operators on the Banach space with the usual linear operations, composition as multiplication and the operator norm.

Closed, W e E. X = E Ф F where F is also a closed linear has a unique representation 2 Clearly Q = Q and x = y + z , R(Q) = E . yeE, We must show that Let x ->■ x in X and Q(x ) -> w in X. Since E is n n x - Q(x ) e F and x - Q ( x ) - > x - w . Therefore, as n n x - w e F. Consequently we see that w = Q(x). This makes Now F is closed, Q into a closed linear mapping which, by the closed graph theorem [22, p. 3) X onto onto R(T). Inasmuch as is one-to-one and the projection T R(T). P is a one-to-one continuous linear By [22, p.